An Introduction to Compressive Sensing

Speaker: Ying-Jou ChenAdvisor: Jian-Jiun Ding

Compressive

Compressed

SensingSamplingCS

Outline

• Conventional Sampling & Compression• Compressive Sensing• Why it is useful?• Framework• When and how to use• Recovery• Simple demo

Review…

Sampling and Compression

Nyquist’s Rate

• Perfect recovery

Transform Coding

• Assume: signal is sparse in some domain…• e.g. JPEG, JPEG2000, MPEG…1. Sample with frequency . Get signal of length N2. Transform signal K (<< N) nonzero

coefficients3. Preserve K coefficients and their locations

Compressive Sensing

Compressive Sensing

• Sample with rate lower than !!

• Can be recovered PERFECTLY!

Comparison

Nyquist’s Sampling Compressive Sensing

Sampling Frequency

Recovery Low pass filter Convex Optimization

Some Applications

• ECG• One-pixel Camera• Medical Imaging: MRI

Framework

Φ¿𝑦 𝑓

NM

N

M

N: length for signal sampled with Nyquist’s rateM: length for signal with lower rate Sampling matrix

𝐲=𝚽𝐟

When? How?

Two things you must know…

When….

• Signal is compressible, sparse…

Ψ

𝑥Φ¿

𝑦 𝑓

NM

N

M

Example… ECG

: 心電圖訊號: DCT (discrete cosine transform)

How…

• How to design the sampling matrix?• How to decide the sampling rate

(M)?

Ψ

𝑥Φ¿

𝑦M

N

Sampling Matrix

• Low coherence

Ψ

𝑥Φ¿

𝑦 Low coherence

Coherence

• Describe similarity

– High coherence more similar Low coherence more different

Example: Time and Frequency

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

• For example, • ,

Fortunately…

• Random Sampling– iid Gaussian N(0,1)–Random

• Low coherence with deterministic basis.

More about low coherence…

Random Sampling

Sampling Rate

• Can be exactly recovered with high probability.

Theorem

C : constant S: sparsityn: signal length

Recovery

Ψ

𝑥Φ¿

𝑦

BUT….

M

N

M

N

𝑓

N

Recovery

• Many related research…– GPSR (Gradient projection for sparse reconstruction)– L1-magic– SparseLab– BOA (Bound optimization approach)…..

Total Procedure

f Find an incoherent matrix e.g. random matrix

Sample signal

𝒂𝒓𝒈 �̂�𝒎𝒊𝒏‖�̂�‖𝟏𝑠 . 𝑡 . 𝐲=𝚯�̂� �̂�=𝐇�̂�

Sampling (Assume f is spare somewhere)

Recovering

已知 :

Demo Time

Reference• Candes, E. J. and M. B. Wakin (2008). "An Introduction To Compressive Sampling."

Signal Processing Magazine, IEEE 25(2): 21-30.• Baraniuk, R. (2008). Compressive sensing. Information Sciences and Systems, 2008.

CISS 2008. 42nd Annual Conference on.• Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An

Introduction to Compressive Sensing.• https://sites.google.com/site/igorcarron2/cs#sparse• http://videolectures.net/mlss09us_candes_ocsssrl1m/

Thanks a lot!